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$k$ is a field. Let $X$ be a connected pointed CW-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

1. $r\circ i= id$
2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
3. and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

$k$ is a field. Let $X$ be a connected pointed $CW$-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

1. $r\circ i= id$
2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
3. and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

$k$ is a field. Let $X$ be a connected pointed CW-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

1. $r\circ i= id$
2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
3. and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

`

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional  ? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

`

$k$ is a field. Let $X$ be a connected pointed CW-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

1. $r\circ i= id$
2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
3. and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

$k$ is a field. Let $X$ be a connected pointed CW-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

1. $r\circ i= id$
2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.

`

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional ? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

`

$k$ is a field. Let $X$ be a connected pointed CW-complex such that the homology $H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous mapping $$r: X\rightarrow K(\pi_{1}(X),1) $$ from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping $$ i: K(\pi_{1}(X),1) \rightarrow X $$ such that:

1. $r\circ i= id$
2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
3. and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

`

My question is the following: Is the homology of $\tilde{X}$ (the universal covering of $X$) finite dimensional ? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$?

`